At each time \(t_i\) at which more than one of the subjects has an event, let \(d_i\) be the number of events at that time, \(D_i\) the set of subjects with events at that time, and let \(s_i\) be a covariate vector for an artificial subject obtained by adding up the covariate values for the subjects with an event at time \(t_i\). Let \[\bar\eta_i = \beta_1s_{i1}+\cdots+\beta_ps_{ip}\] and \(\bar\theta_i = \text{exp}{\left\{\bar\eta_i\right\}}\).
Let \(s_i\) be a covariate vector for an artificial subject obtained by adding up the covariate values for the subjects with an event at time \(t_i\). Note that
\[
\begin{aligned}
\bar\eta_i &=\sum_{j \in D_i}\beta_1x_{j1}+\cdots+\beta_px_{jp}\\
&= \beta_1s_{i1}+\cdots+\beta_ps_{ip}\\
\bar\theta_i &= \text{exp}{\left\{\bar\eta_i\right\}}\\
&= \prod_{j \in D_i}\theta_i
\end{aligned}
\]
Breslow’s method for ties
Breslow’s method estimates the partial likelihood as
\[
\begin{aligned}
L(\beta|T) &=
\prod_i \frac{\bar\theta_i}{[\sum_{k \in R(t_i)} \theta_k]^{d_i}}\\
&= \prod_i \prod_{j \in D_i}\frac{\theta_j}{\sum_{k \in R(t_i)} \theta_k}
\end{aligned}
\]
This method is equivalent to treating each event as distinct and using the non-ties formula. It works best when the number of ties is small. It is the default in many statistical packages, including PROC PHREG in SAS.
Efron’s method for ties
The other common method is Efron’s, which is the default in R.
\[L(\beta|T)=
\prod_i \frac{\bar\theta_i}{\prod_{j=1}^{d_i}[\sum_{k \in R(t_i)} \theta_k-\frac{j-1}{d_i}\sum_{k \in D_i} \theta_k]}\] This is closer to the exact discrete partial likelihood when there are many ties.
The third option in R (and an option also in SAS as discrete) is the “exact” method, which is the same one used for matched logistic regression.
Example: Breslow’s method
Suppose as an example we have a time \(t\) where there are 20 individuals at risk and three failures. Let the three individuals have risk parameters \(\theta_1, \theta_2, \theta_3\) and let the sum of the risk parameters of the remaining 17 individuals be \(\theta_R\). Then the factor in the partial likelihood at time \(t\) using Breslow’s method is
\[
\left(\frac{\theta_1}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_2}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_3}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\]
If on the other hand, they had died in the order 1,2, 3, then the contribution to the partial likelihood would be:
\[
\left(\frac{\theta_1}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_2}{\theta_R+\theta_2+\theta_3}\right)
\left(\frac{\theta_3}{\theta_R+\theta_3}\right)
\]
as the risk set got smaller with each failure. The exact method roughly averages the results for the six possible orderings of the failures.
Example: Efron’s method
But we don’t know the order they failed in, so instead of reducing the denominator by one risk coefficient each time, we reduce it by the same fraction. This is Efron’s method.
\[\left(\frac{\theta_1}{\theta_R+\theta_1+\theta_2+\theta_3}\right)
\left(\frac{\theta_2}{\theta_R+2(\theta_1+\theta_2+\theta_3)/3}\right)
\left(\frac{\theta_3}{\theta_R+(\theta_1+\theta_2+\theta_3)/3}\right)\]